Fractals and ChaosMaster of School Mathematics
Introduction – Part 1
After reviewing the last eleven years of Mathematics Teacher magazines, I decided to
focus on two closely related topics that I knew little about. They are fractals and chaos. Since, I
am also beginning a new position this year, the activities I have chosen for my classroom are not
meant to be stand-alone units. Their purpose is to supplement current material and to connect
with past lessons.
What are fractals? Benoit Mandelbrot created the term fractal in the 1970s. Fractals are
defined as “objects that appear to be broken into many pieces, each piece being a copy of the
entire shape” (Naylor, 1999, 360). The name was actually given because fractals exist in
fractional dimensions. To visualize a fractal, consider a head of cauliflower or a bunch of
broccoli. If a piece is broken off from either of these, the part still resembles the whole. This is
the fractal property of self-similarity. In nature, examples of fractals are mountains, coastlines,
and rivers.
The fractals closely related cousin is chaos. An illustration of mathematical chaos is
Edward Lorenz’s experiment with convection currents (Crownover, 1995). Using the same
model, vastly different answers were obtained when Lorenz introduced truncated values halfway
through a run. He observed a characteristic of chaos, “sensitive dependence on initial
conditions” (130). In layman’s terms, Lorenz referred to it as the “butterfly effect” which
questions whether “the flap of a butterfly’s wings in Brazil set off a tornado in Texas” (130).
Fractals have an extensive history in spite of the fact that the word fractal was not
introduced until the 1970s. For instance, the Cantor Set (1872), Sierpinski Gasket (1916), and
Koch Curve (1904) have been known for quite some time (Peitgen, Jurgens, and Saupe, 1992).
However, they were viewed as mathematical monsters. They were used “to demonstrate the
deviation from the familiar rather than to typify the normal” (76). For example, the Koch Curve
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is familiar to many people because of its snowflake shape. To mathematicians, nevertheless, the
Koch Curve was seen as a monster, because its boundary is continuous but not differentiable. As
a twenty-year-old student in the 1940s, Mandelbrot recalls being told by a professor “as a topic
of active research, the geometry I loved was dead” (9). Furthermore, “it had been dead for
nearly a century except in mathematics for children” (10).
In Mandelbrot’s opinion, the turning point in fractal study occurred in 1979-1980 with his
research of the Fatou-Julia theory of iteration (Peitgen, Jurgens, and Saupe, 1992). This theory
had last been changed in 1918. Mandelbrot used a computer to investigate a small portion of
Fatou-Julia, which he referred to as the μ-map. It was later renamed the Mandelbrot Set (M-Set)
in his honor by Adrien Douady and John Hubbard. Then, in 1983, Mandelbrot published The
Fractal Geometry of Nature in which he connected the mathematical monsters of old together
into the category of fractal geometry (Camp, 2000).
Mandelbrot credits computers with the awakening of “experimental mathematics”
(Peitgen, Jurgens, and Saupe, 1992, 9). The computer not only allowed for enhanced
calculations but also provided graphics for previously abstract ideas. As the 1970s progressed,
Mandelbrot realized mathematics could be studied using the computer with the same methods he
had been applying to physics. It is likely no coincidence that Mandelbrot chose to revisit Fatou-
Julia. His uncle, a mathematics professor in France, had provided a twenty year old Mandelbrot
with the original reprints of the theory in 1945 in an unsuccessful attempt to try to influence the
directions of his nephew’s career. Much to the dismay of the uncle, computers, not pure
mathematics, led to the discovery of the Mandelbrot Set. In chaos theory, Lorenz also used
computers in the 1961 study of convection currents. He developed a system of twelve ordinary
differential equations to simulate the atmosphere. Computers were than used to solve the
equations and the data was examined to determine the validity of the model. The chaos finding
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ultimately led to Lorenz’s presentation of the “butterfly effect” in 1979 before the American
Association of Advancement of Science in which he discussed the impossibility of long-term
weather forecasts (Crownover, 1995). The study of fractals and chaos in research and at the
collegiate level has been fueled by the advent of the computer. At the secondary mathematics
level, graphing calculators lead the charge because of their low cost and ease of portability.
Fractals and chaos are two topics that are very capable of maintaining students’ interests.
This is at least partially due to their variety of uses. They are used to study the spread of forest
fires and epidemics (Camp, 2000). A fractal process was used to compress images for
Microsoft’s Encarta CD-ROM encyclopedia. Richard Voss is currently investigating “how
automated fractal interpretation of satellite images can help monitor the health of the
environment” (711). Fractals have been used to produce footage for films such as Dante’s Peak
and Star Trek II: The Wrath of Khan (Camp, 2000; Crownover, 1995). Fractals are additionally
being studied to see if they can be used to identify cancerous tumors (Camp, 2000). Visually,
fractals are easier for students to understand than Euclidean geometry. At the same time, fractals
and chaos seem to be tailor-made for today’s technologically inclined students.
Literature Review
In an article on increasing levels of mathematical maturity, Hare and Philliby give six
characteristics of mathematically mature calculus students. Three of the six are relevant here and
are listed below.
concepts from multiple perspectives.
Characteristic 2: The student has a strong understanding of prerequisite algebra
concepts that connect with calculus concepts.
Characteristic 3: The student is able to use technology (2004, 7).
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The study of mathematics from multiple perspectives, an understanding of prerequisites, and the
use of technology are prevalent throughout the remainder of this section.
A prerequisite for calculus is the understanding of functions. Clement (2001) states,
“although the function concept is a central one in mathematics, many research studies of high
school students have shown that it is also one of the most difficult for students to understand”
(745). Consequently, researchers now advocate approaching functions through multiple
representations. These are algebraic, numeric, and graphic (Cunningham 2005; Beckman,
Thompson, and Senk, 1999; Keller and Hirsch, 1998). According to Robert Cunningham, there
are six types of transfers between representations. They are illustrated below.
Cunningham found that students have difficulty changing between representations to solve
problems. Most troubling of all are transfers involving graphs. Six transfer problem types from
his study follow.
A = Algebraic N = Numeric G = Graphic
A to G Graph the linear equation 3x + 2y = 12 using slope and y-intercept.
G to A Given the graph shown, determine the linear equation represented. (Assume each tick mark represents one unit on the graph)
A to N Given the linear equation 3x + 2y = 12, construct a table displaying ordered pairs that are solutions to the equation.
N to A Given a set of two ordered pairs, determine the equation of the line that passes through them.
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N to G Given a set of ordered pairs, graph the linear equation that passes through them.
G to N Given the graph shown, determine an ordered pair that satisfies the linear equation. (Assume each tick mark represents one unit on the graph)
(Cunningham, 2005, 79)
One such study that Cunningham (2005) reviews involving functions and transfers was
completed in 1999 by Porzio. In the study, Porzio compared three sections of calculus. Their
approaches included: traditional, graphing calculator, and the use of Mathematica. The
traditional and graphing calculator sections performed poorly on transfer problems such as the
following exercise from their posttest.
The population of a herd of deer is given by the function P(t) = 4000 – 500cos2πt where t is measured in years and t = 0 corresponds to January 1.
a) When in the year is the population at its maximum? What is that maximum? b) When in the year is the population at its minimum? What is that minimum? c) When in the year is the population increasing the fastest? d) Approximately how fast is the population changing on the first of July?
(Porzio, 1999, 11) Porzio found that the Mathematica group did much better since “they had been given many
opportunities to solve transfer problems” (Cunningham, 2005, 74). The graphing calculator class
had seen multiple representations modeled. However, they had not been made to practice
transferring on their own. Porzio’s conclusion, which has been duplicated several times since
1999, is that teachers must require students to solve problems and make connections using
multiple representations (Cunningham 2005; Beckman, Thompson, and Senk, 1999).
Demonstration alone is not effective in building an understanding of functions. In part three, of
this paper, graphing calculators will be utilized in high school mathematics classes as part of a
multi-representational study of functions related to fractals.
Another important idea in secondary mathematics is algorithms. According to Mingus
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and Grassl (1998), the “dominance of algorithmic procedures has caused many people to define
knowledge in mathematics as competence in executing prescribed algorithms” (32). Usiskin
(1998) breaks this down further by classifying the typical algorithms used in school mathematics
into three categories. They are arithmetic algorithms, algebra and calculus algorithms, and
drawing algorithms such as graphing and geometric constructions. Nevertheless, “current reform
movements are de-emphasizing the importance of algorithms in favor of problem-solving
approaches, the conceptualization of mathematical processes, and applications of mathematics in
real-world situations” (Mingus and Grassl, 1998, 32). This is carried out primarily through
algorithmic and recursive thinking. Polya’s four-step model is cited as an example of
algorithmic thinking. It is summarized as understand the problem; devise a plan; implement the
plan; extend the problem. Recursive thinking, on the other hand, is distinguished as iterative,
self-referential, and never ending. Students need to have the chance to implement both types of
thinking in creating their own rules to describe patterns and relationships that are found in
problem solving (Curio and Schwartz, 1998). In section three of this paper, students will have
the opportunity to practice both types of thinking as they examine the recursion found in fractals
and chaos and model population growth.
A somewhat related approach to the reform movement cited above is a proposal to follow
the work of fractal geometers (Simmt, 1998). “Rather than use algorithms simply to determine a
result, these mathematicians use computer algorithms to generate objects, such as the Mandelbrot
Set, that are of mathematical interest” (274). Simmt continues on to explain that the object then
becomes the focus of investigation. By applying algorithms in this manner, students have a
chance to work with fractal geometry and iteration. It also allows students to make connections
between topics such as “measurement, sequences and series, geometry and limits” (277). Simmt
is not alone, however, in acknowledging the continued necessity to use algorithms in traditional
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ways (Curio and Schwartz, 1998).
Recursion is presented in the preceding paragraphs as an alternative approach to teaching
algorithms. The standard notation used in the recursive process can confuse students (Hart,
1998; Franzblau and Warner, 2001). In Franzblau and Warner’s study students were given the
recursive formula Fn = Fn-1 + 3 (192).
Several misinterpretations were documented. Some students thought Fn-1 “meant take the answer
and subtract 1” rather than use the previous answer (191). These same students believed Fn+1
meant add one to the answer. Yet, others confused Fn+1 with its functional counterpart F(n+1).
Mistakes were also attributed to sloppiness after the students were interviewed.
There are two primary schools of thought on how to deal with this issue. Hart (1998)
claims that the formal subscript notation can prevent students from understanding basic
recursion. He recommends replacing subscript notation with “NOW and NEXT” until “students
have a solid conceptual understanding of recursion” (264,265). Nevertheless, Franzblau and
Warner (2001) are proponents of using formal subscript notation from the start. Their study
shows that the use of “ANSWER and PREVIOUS or NEXT and NOW may actually be a
barrier” to learning standard notation (197). Furthermore, students who do not understand the
notation generally ignore it until they feel comfortable with it. Franzblau and Warner have two
recommendations for teachers. Model formal subscript notation from the start, but do not
initially focus on it. Secondly, they found that students who could effectively describe patterns
in words were most likely to use subscript notation immediately. Hence, educators should
require students to write pattern descriptions. There is one similarity present in the above
research. Hart (1998) and Franzblau and Warner (2001) agree that learning formal subscript
notation is a long-term process regardless of the method that is chosen. Nonetheless, there is no
clear answer to teaching recursion’s standard notation.
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The majority of the research presented to this point leads directly to the Algebra Standard
found in the Principles and Standards for School Mathematics (2000). It states:
Instructional programs from prekindergarten through grade 12 should enable
students to-
• represent and analyze mathematical situations and structures using
algebraic symbols;
relationships;
• analyze change in various contexts (32).
Technology may be able to assist even more in the implementation of the aforementioned
standard. President of the NCTM, Cathy Seeley (2004), says that graphing calculators have
become “essential components of an effective mathematics program” that students need to be
able to use (25). She does not address the issue of programming.
Mayer, Dyck, and Vilbers (1986) do address programming in “Learning to Program and
Learning to Think: What’s the Connection?”. They conducted a study in which 111 computer
naïve college students took part. Neither group consisted of engineering or programming
majors. Fifty-seven students were taught BASIC. After the pretest and posttest scores were
analyzed, the BASIC group gained substantially more on two skills: “problem translation (as
measured by word problem translation and word problem solution) and procedure
comprehension (as measured by following procedures and following directions)” (607). Mayer,
Dyck, and Vilbers conclude that programming “can have positive effects on thinking skills that
are directly related to the language to be learned” (609).
Today, programming can be done on graphing calculators. In Micromath, Alan Grahm
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(2003) states “ the graphics calculator’s programming language is essentially BASIC” (41). He
has found that “using algebra is almost unavoidable when writing even a simple program” (41).
Consequently, students both learn and apply algebra in the context of programming.
One final piece of research examines the four mathematical learning styles that have been
identified (Strong, Thomas, Perini, and Silver; 2004). Students who are best characterized as
having the mastery style benefit most from repetitive practice and working step-by step. Other
students are categorized with an understanding style and view mathematics conceptually while
looking for the reasoning behind it. Still, students with interpersonal styles thrive on application
and cooperative learning. Yet, students with self-expressive styles are problem solvers who need
to investigate and visualize. Therefore, teachers should attempt to accommodate the above styles
by using a variety of strategies that work both to the strengths of students while forcing them to
improve on their weaknesses. In part three, these styles will be reconciled by approaching fractal
related activities algebraically, numerically, and graphically on both an individual and
cooperative basis that involve repetitive practice, problem solving, and application.
The reviewed literature strongly supports approaching the study of mathematics from
multiple perspectives. Educators must utilize algebraic, numeric, and graphic techniques in
order to make the study of functions more meaningful. At the same time, algorithmic procedures
must be de-emphasized while algorithmic and recursive thinking is applied to problem solving.
Technology, namely graphing calculators at the high school level, may be able to help in
achieving the above goals. In fact, technological proficiency is itself a standard to be reached
prior to students being considered mathematically mature. Only after these steps are taken will
students enter calculus with the prerequisite knowledge necessary to be successful.
In the next section, the Mandelbrot Set is a by-product of the technological advances that
have taken place within the last fifty years. The computer’s ability to manipulate the large
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amount of data necessary to produce the recursively-defined fractal is a recent development. The
presentation of the Mandelbrot Set will remain true to the previously cited research in that it will
be investigated algebraically, numerically, and graphically.
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Benoit Mandelbrot, the father of fractal geometry, discovered a fractal later to be known
as the Mandelbrot Set (M-Set) in 1980. From all indications, the M-Set has been studied
intensively ever since. One text, Peak and Frame’s Chaos Under Control: The Art and Science
of Complexity, introduced topics related to chaos and fractals for undergraduate mathematics
students and opened the door to the realm of possible applications for the M-Set in secondary
mathematics.
The Mandelbrot Set is pictured below. First and foremost, it is a fractal. The
<http://fract.ygingras.net/grey/-0.65000/0.00000/0.25 >
property of self-similarity is exhibited on the boundary where many smaller copies of the M-Set
are found, although, they are somewhat distorted. Distortion is common among fractals that
occur in nature, which lack the manufactured precision of other fractals such as Sierpinski’s
Triangle or the Cantor Dust. Nevertheless, this lack of precision is far outweighed by the
uniqueness of the M-Set’s Elephant and Seahorse Valleys. These are named for the shapes that
are seen upon enlargement. While the physical appearance of the M-Set may be impressive, it is
nothing compared to the mathematics from which it is generated.
“The Mandelbrot Set µ for the quadratic fc(z) = z2 + c is defined as the collection of all
c C (where C is the complex plane) for which the orbit of the point 0 is bounded, that is,
µ = { c C :{ fc (n) (0)} ∞ n=0 is bounded}”
(Crownover, 1995, 209).
More specifically, the distance between a point and the origin may be no greater than two in
order to be included in the M-Set.
Theorem “If | c | > 2 and | z | ≥ | c |, then the orbit of z escapes to ∞.”
(210-11)
Proof:
| zn 2 | = | zn
2 + c - c |
| zn 2 | ≤ | zn
2 + c | + | c |
| zn 2 + c | ≥ | zn | ( | c | - 1) | zn | ≥ | c |
| zn 2 + c | ≥ | zn | ( 2 + p – 1 ) | c | = 2 + p
| zn 2 + c | ≥ | zn | ( p + 1 )
Iterating | zn+1 | ≥ | zn | ( p + 1 )
| zn+2 | ≥ | | zn | ( p + 1 ) | ( p + 1 ) = | zn | ( p + 1 )2 p > 0
| zn+3 | ≥ | | zn | ( p + 1 )2 | ( p + 1 ) = | zn | ( p + 1 )3 p > 0
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| zn+k | ≥ | zn | ( p + 1 )k
Therefore, the orbit of z approaches infinity as n approaches infinity (Crownover, 1995).
The Mandelbrot Set may also be generated in the real number system by
xn+1 = xn 2 – yn
2 + a and
yn+1 = 2xnyn + b
where x0 = y0 =0 with an orbit of 0. This is essentially the same as the complex definition.
Consider the complex z-plane. The real z are on the x-axis while the imaginary are on the y-axis.
So for any complex number
z = x + yi
zn = xn +yni
zn+1 = (xn +yni)2 + a + bi
= xn 2 + 2ixnyn + i2 yn
2 + a + bi
= xn 2 – yn
(Frame and Peak, 1994)
Hence, the above two methods of generating the M-Set will be used interchangeably throughout
the rest of this paper.
The mathematics involved in the M-Set is also readily apparent in a more concrete
manner. For example, periodic cycles are found throughout the M-Set. These cycles are defined
such that for some p>0, x0 = xp (Peitgen, Jurgens, and Saupe, 1992). In other words, if a bud is
a 4-cycle (p=4), the values will repeat after four complete iterations. “The antenna structure
attached to each bud attached to the main cardioid codes the period of the attractor to which the
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orbit of 0 converges for any (a,b)- value taken from the body of that bud” (Frame and Peak,
1994, 253). This is illustrated in Figure 1 below. Note how the bud labeled D has the numbers
Figure 1: Mandelbrot Set with cycles
<http://fract.ygingras.net/grey/-0.65000/0.00000/0.25>
1, 2, and 3 near its antennas. This is one way of knowing the bud is a 3-cycle. Later, the
calculations that support D will additionally show a repeating pattern of groups of three.
Another item that is evident is that the regions labeled D, E, and F appear twice. That is because
the horizontal line (a-axis) that goes through the cleft of the cardioid serves as the line of
symmetry. The portion of the M-Set above this line is known as the Northern Principal
Sequence while the portion below it is the Southern Principal Sequence. The
corresponding buds in both sequences have the same cycle.
In Appendix A, six points are used to illustrate the cyclic behavior of the M-Set. Using
the definition from the real number system,
xn+1 = xn 2 – yn
2 + a and
yn+1 = 2xnyn + b
where x0 = y0 =0, Microsoft Excel was used to find twenty-five thousand iterations of each point.
The final thirty iterations are included in Appendix A. The point A was found using a = .25 and
b = 0. (Note: The (a,b) chosen for point A and the following Excel calculations represent a real
ordered pair (x,y), not the complex number c). The a, b, and distance columns converge to .5, 0,
and .5 respectively. Since this occurs, it is said that (.25,0) with an orbit of 0 (x0 = y0 =0)
converges to a fixed point. The intercept, (.25,0) was chosen because of its convenience in
illustrating the proof of convergence.
xn+1 = xn 2 – yn
2 + a = xn 2 – 02 + .25 = xn
2 + .25
(.25,0) converges to a fixed point.
xn+1 = xn and yn+1 = yn
xn+1 = xn 2 + ¼
xn = xn 2 + ¼ by substitution
0 = xn 2 - xn + ¼
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xn = ½
The solution is said to attract or converge to a fixed point if | derivative at the fixed point | ≤
1and to repel if | derivative at the fixed point | > 1.
If xn+1 = xn 2 + ¼
Let f(x) = x2 + ¼
f’(x) = 2x
f’ (1/2) = 2(1/2) = 1 Therefore, x = ½ is a fixed point that attracts.
The next point, B, is located in the head of the M-Set. The list of iterations shows that
approximately the same two values keep repeating in each column. Thus, each point in the head
is a 2-cycle. A similar repeating pattern is also seen on the Excel calculation pages for points C,
D, E, F, and G. (Additional convergence proofs are also included for B, C, and G.) However, C
and D are 3-cycles, E and G are 4-cycles, and F is a 5-cycle. Note that the (a,b) of D, E, and F
were deliberately chosen by magnifying the Mandelbrot Set at <http://www.softlab.ece.ntua.gr/
miscellaneous/mandel/mandel.html>. Although the cyclic behavior of E and F can still be found
by counting the antennas in Figure 1 as previously stated, this method is severely limited by the
clarity of the given diagram. Iterations, on the other hand, will always work. Furthermore,
computers and graphing calculators (Appendix A) can be programmed to handle the necessary
calculations.
The Farey Sequence
The Farey sequence is another method by which the periodic cycle of a bud may be
found. The only significant constraint on this sequence is that the two starting buds must have
consecutive cycles. The consecutive cycles of the Northern Principal Sequence (NPS) are
labeled on the next page. Informally, the Farey sequence states the smallest periodic cycle
between two consecutive buds may be calculated by summing their cycles. More specifically,
“two given buds of periods p and q at the cardioid determine the period of the largest bud in
between them as p + q” (Peitgen, Jurgens, and Saupe, 1992, 439). These seemingly
contradictory definitions are reconciled when one understands the cycle size and physical size of
a bud are inversely related. Examination of the NPS above shows the physical size of a bud
decreases as the cycle increases.
Recall that B is a 2-cycle and D is a 3-cycle. Scrutinize the region between B and D
either on the top half of the M-Set or the bottom half. According to the Farey sequence, the
smallest cycle between B and D is 2 + 3 = 5. This has been previously identified as letter F. In a
manner of speaking, the Farey sequence works similar to Pascal’s Triangle. Consider the
following diagram and associated picture (for visual reference only). To find the smallest cycle
of a bud between B and F, add the two and the five to get seven. This pattern is also found
between letters D and E. Remember D is a 3-cycle and E is a 4-cycle.
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Thus, the smallest cycle that can be found is 3 + 4 = 7. The Farey sequence may also be
applied to the buds on the buds.
Each bud on the Mandelbrot Set additionally contributes a unique arrangement of its
own. First of all, recall that the Northern Principal Sequence (NPS) is made up of buds from left
to right of consecutive cycle sizes 2 (B), 3 (D), 4 (E), 5, 6, 7, and so forth. The head, B, is a
2-cycle and its largest bud is found on the left side, which is a 4-cycle. Proceeding either
clockwise or counterclockwise, the next smaller bud is a 6-cycle. The next smaller bud is an 8-
cycle and so on. These cycles are obtained by multiplying the cycle size of the head by the
consecutive cycle sizes from the cardioid (Peak and Frame, 1994).
2 · 2 2 · 3 2 · 4 2 · 5 ····
4 6 8 10 ····
Pictured at left: Head
Pictured above: D
The same idea should be applied to the 3-cycle D pictured above. Its largest bud is found at the
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top (NPS). This is the dividing point. It is also a 6-cycle. Proceeding either clockwise or
counterclockwise in the same fashion as with the head yields cycles of the following sizes.
3 · 2 3 · 3 3 · 4 3 · 5 ····
6 9 12 15 ····
This pattern generalizes to work for all buds. The 6-cycle bud attached at the top of D has buds
of sizes
12 15 24 30 ····
The Farey sequence may then be applied to show that the smallest possible cycle or period of the
largest bud that exists between the 12 and 15-cycles above is a 27 cycle.
Bifurcation Diagrams and Logistic Functions
The cyclic behavior of the Mandelbrot Set is also seen in its bifurcation diagram.
Bifurcation diagrams are used to give a “summary of the dynamics of the entire family”
including the “transition to chaos” (Devaney, 1992, 82). In this case, the M-Set’s bifurcation
diagram is actually a transformed version of the logistic function’s bifurcation map. Recall the
M-Set is defined by
2 + a and
x’n+1 = sx’n (1-x’n).
where “ ’ ” is used because no connection has yet been established between the maps (Frame and
Peak, 1994). If b = 0 and yn = 0 then
xn+1 = xn 2 + a
A transformation connecting xn to x’n is needed. Let
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Substitute and solve for x’n+1.
Ax’n+1 + B = (Ax’n + B)2 + a
Ax’n+1 + B = A2x’2 n + 2ABxn’ + B2 + a
Ax’n+1 = A2x’2 n + 2ABxn’ + B2 – B + a
(2) x’n+1 = Ax’2 n + 2Bxn’ + (B2 – B + a)/A
Simplify the logistic function.
n
n + sxn’
A = -s and
2B = s so
B = s/2 and
(5) xn = -sx’n + s/2 (Frame and Peak, 1994).
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Therefore, the bifurcation maps of the M-Set and logistic function are the same if equations (4)
and (5) are true.
The bifurcation diagrams of the Mandelbrot Set and the logistic function are found in
Figures 2 and 3. The diagrams illustrate the characteristic of period doubling. In Figure 2,
several M-Set intervals that correspond to the bifurcation diagram are marked with their
associated cycles. The 1-cycle is the fixed point behavior presented earlier. Starting at c = .25,
there are cycles or periods of sizes 1, 2, 4, and a glimpse of 8 before the midget (3-cycle) on the
tendril is seen. These cycles correspond to the number of branches found in the bifurcation
diagram in Figure 2. This period doubling behavior is readily apparent in Figure 3 simply by
counting the branches. The similarities between the M-Set and logistics function continue as
convergence is examined.
At left Figure 2: M-Set Bifurcation Above Figure 3: Logistic Bifurcation from http://enwikipedia.org/wiki/Bifurcation_diagram
Feigenbaum Constant
The bifurcation diagrams may be used to find the Feigenbaum constant, δ. This constant
is defined as the limit of “the ratios of the lengths of successive intervals between bifurcation
points” where as the number of bifurcations, n, approach infinity
δ = lim [(cn – cn-1) / (cn+1 - cn)] ≈ 4.669162 n→∞
(Crownover, 1994, 142).
Because of the difficulty in finding the bifurcation points, cn , the Feigenbaum constant is often
given by
n)] n→∞
(143).
The point c* n may be found between any pair of consecutive bifurcation points cn and cn+1
“where there is a period 2n orbit that is superattracting” (143). A point is said to be
superattracting if its numerical derivative equals zero. These superattracting points may then be
calculated using Newton’s Method. The resulting Feigenbaum constant allows for predictions of
future bifurcations. The logistic function and M-Set do, however, converge on this constant at
different points.
The place at which the Feigenbaum constant is found in Figures 2 and 3 is referred to as
the Feigenbaum point. On the logistics bifurcation diagram, this is reached at approximately
x = 3.57. On the M-Set, the Feigenbaum point is approximately c = -1.401155 (Crownover,
1994). To the right of this c value, period doubling takes place. To the left of it, chaos reigns.
The 3-cycle midget is located in that chaotic region. Moreover, a similar split is present in
Figure 3 around 3.57. The only difference is that period doubling is to the left of the
Feigenbaum point as chaos takes over to the right of it. The bifurcation diagrams and long-term
behavior reveal several common characteristics between the M-Set and logistic function.
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The Cardioid
A unique feature of the Mandelbrot Set is the cardioid shape when graphed in polar form.
The following provides the derivation of the explicit formula for the cardioid’s boundary.
z = z2 + c c is Complex
z' = 2z
Case 1 Case 2 Case 3
| 2z | > 1 | 2z | < 1 | 2z | = 1
fixed point repels fixed point attracts fixed point is neutral
points are outside points are inside points make up boundary
of cardioid of cardioid of cardioid
since | 2z | = 1
since z = z2 + c
z2 – z + c = 0
¼e2iθ-½eiθ + c = 0
c = ½eiθ - ¼e2iθ complex equation of cardioid
By substitution c = ½(cosθ + isinθ) - ¼(cos2θ + isin2θ)
c = ½cosθ + ½isinθ - ¼cos2θ – ¼isin2θ
c = ½cosθ - ¼cos2θ + (½sinθ – ¼sin2θ)i
c = x + yi
y = ½sinθ – ¼sin2θ
(Devaney, 1992; Peitgen, Jurgens, and Saupe, 1992)
26
For each θ, 0 ≤ θ ≤ 2, these two equations produce a point on the boundary of the cardioid.
On the whole, the cardioid’s boundary is yet another piece of the Mandelbrot puzzle.
This puzzle is traditionally colored black because that is how the M-Set was first viewed. The
points that escape to infinity are usually color-coded according to their rate of escape. These
colorful pictures can be found in abundance on the internet. In fact, those pictures are what
originally provided the motivation to choose the Mandelbrot Set as a topic of study. See
Appendix A for an example.
At its most basic level, the Mandelbrot Set is a fractal formed through the process of
iteration. In the next section, students will study three less complex fractals constructed in a
similar manner. Students will also study Newton’s Method, an iterative procedure used to find
the solution of an equation. Furthermore, chaos exhibited possibly best in the M-Set’s
bifurcation diagram will be illustrated in both the Newton’s Method and population growth
activities. The population growth activity will additionally be found to have its origins in the
same logistic function that played such an integral role in the Mandelbrot Set discussion. Lastly,
the research supported approach of studying mathematics from multiple perspectives that was
used to examine the Mandelbrot Set will be implemented in the student activities.
27
The problem solving technique, involving algorithmic and recursive thinking, applied to
the Mandelbrot Set is utilized in the classroom in this section. This is achieved through a series
of lessons that are split into three activities. The first has the overall purpose of beginning the
school year using fractals to introduce the concept of limits in calculus. The second activity
should occur during the second quarter of calculus when the connection between Newton’s
Method and chaos is possible. The last activity is for use in pre-calculus during the second
quarter. Logistic functions will be used to model population growth with a twist that involves
chaos.
The ideas behind the three activities come from the Mathematics Teacher. The limit
lessons are a compilation of several articles from geometry reconstructed to lead into calculus.
The remaining activities have primary sources. The second activity is based upon “Black Dot:
Newton’s Method and a Simple One-Dimensional Fractal” by Tony J. Fischer. Meanwhile, the
activity involving logistic functions closely resembles “Chaotic Behavior in the Classroom” by
Robert C. Iovinelli. All the activities are approached graphically, numerically, and analytically.
Furthermore, each also contains a technology component.
Limits – Calculus – Appendix B
The limits activity studies what happens to three fractals as they grow through the process
of iteration. Students are to analyze patterns from the fractal tree and Cantor Set (Dust) before
limit notation is formally introduced. The Sierpinski Gasket (Triangle) is also a fractal.
Although, it is going to be approached from a different perspective – programming. After the
SIERPINS program has been discussed, the students enter it into their graphing calculators. For
many of the students, this is expected to be their first programming experience.
29
Immediately prior to the start of the school year, the decision to postpone the limits
activity was made. The decision was based on the fact that the teacher has no previous
experience with her incoming students, because she is new to the district. As a result, a one
week intensive review of graphing relations was undertaken both with and without graphing
calculators. Topics such as x and y intercepts, maximums, minimums, and piecewise functions
were discussed. This is the assumed background with which the activity began.
The first day started with the students breaking into small groups to work on the fractal
tree and Cantor Set worksheets. It quickly became apparent that the students possessed no
previous knowledge of fractals, which was evident by the initial conversations that took place in
their cooperative groups. It was clear the term fractal held no meaning. As a result, after the
fractal tree was drawn, the class reconvened briefly as a whole to discuss fractal growth and self-
similarity. The students then returned to small group work. Clarification was needed as to what
constitutes iteration 0. Students also requested a hint on generalizing the information organized
in the tables. Consequently, the entire class examined the relationship between the number of
iterations and the number of new branches (5a of fractal tree) before returning to group work.
The period ended with the biggest problem having been a typographical error in which inches
had been inserted instead of units.
The second day revolved around the completion of the fractal tree and Cantor Set lessons.
Student volunteers were used to explain the recursive patterns they had discovered. This
educator provided additional guidance in the discussion that took place over what happens to the
length of a single new branch as n approaches infinity (6 of fractal tree). The students
understood the numerical and graphical aspects of the exercise immediately. Support was given
in the form of the introduction to formal limit notation. These fractals appear to have provided
an intuitive approach to limits. The students did not have difficulty making the transition from
30
the iteration-produced patterns of fractals to formal limit notation. Proof of this successful
transition was observed in the student led discussion of the Cantor Set lesson. The liveliest
exchange of the day was actually an algebraic question on whether 2 2n – 1 = 2n+1 – 1. The day
ended with a brief introduction to programming during which the students accurately guessed the
purpose of about half the steps in the Sierpinski program. They had a particularly high degree of
success analyzing the first half of the program. The class then entered the first few lines of the
program into their calculators.
The final day of the activity was spent working on the Sierpinski Gasket. About half of
the time was used to debug programs. The students’ mistakes varied from missing decimal
points to skipping entire lines of code to typing in commands instead of using calculator defined
features. The students had never programmed before, however, one decided to program the
quadratic formula into his calculator after this experience. The students struggled with the last
two questions of the problem set. The class ended up drawing the first few iterations of the
Sierpinski Gasket by hand before everyone realized the corners will always remain. On the last
question involving the remaining area of the original triangle, the students had to be reminded
that points are zero dimensional before it could be completed.
Connections to Research
In the literature review, the importance of students having the opportunity to apply
algorithmic and recursive thinking in creating rules to describe patterns and relationships is cited.
The students were given the chance to analyze recursive patterns in the fractal tree and Cantor
Set lessons. A deliberate decision was also made to follow Hart’s lead in the handling of the
symbolic notation involved in recursion. This educator wanted students to focus on the new
notation in limits, not become overwhelmed in recursive notation. As a result, the students were
asked if their rules would take them from the first step to the second, from the second to the
31
third, and so forth. Meanwhile, multiple representations were used to facilitate the understanding
of functions. For example, exercise five on the fractal tree lesson and problems two and three on
the Cantor Set lesson were approached numerically and graphically. It was during the discussion
over the Cantor Set that it became evident that not all of the students have the prerequisite
algebra skills necessary to qualify as a mathematically mature calculus student. In fact, one
student remains steadfast in his refusal to acknowledge the validity of the statement 2 2n – 1 =
2n+1 – 1. Finally, the use of graphing calculator technology had two purposes. Graphing
calculators were used in the attempt to approach calculus from multiple perspectives and to lay
the groundwork necessary to program Newton’s Method.
Future
No changes to this activity are planned. It introduced students to fractals, limits, and
programming. Compared to previous classes, these students appear to have a better grasp on the
concept of limit. Even though attitude is not a quantifiable measure, this class approached limits
more willingly. The most pleasant surprise is the ease with which students approached
programming. In the past, when teaching, the first experience the teacher usually presents to
students involves the rectangle approximation method for calculating the area under a curve.
The experience proved difficult at best. There is no clear reason why programming the
Sierpinski Gasket should have been easier. Overall, this activity is considered successful.
Newton’s Method – Calculus – Appendix C
The students are expected to have a much broader knowledge base before starting this
activity. Prior to studying Newton’s Method, the students will have spent close to a month
studying derivatives, concluding with a section on linearizations. They will have additionally
begun looking at uses of the first and second derivative in identifying maxima, minima, and
32
inflection points. However, the students will still have very limited programming skills. These
are the experiences upon which Newton’s Method builds. (Please see Appendix C for an
explanation of terms.)
Newton’s Method is a procedure used to approximate the zeros or solutions of an
equation using a sequence of linearizations. A graphing calculator program will be used to
investigate Newton’s Method in this activity. First and foremost, nevertheless, the students will
work with the algorithm
xn+1 = xn – f(xn)/f’(xn)
by hand to understand how it works before writing a program for their calculators. Since
students know they can get the zeros of many functions immediately by using the root or zero
key on their calculators, they usually do not want to waste time working on the repeated
iterations necessary for Newton’s Method. Therefore, this activity will take a slightly different
approach by engaging students in an experiment involving chaos. The final product will be a
number line colored according to the zeros on which the initial guesses converge.
On the first day of this activity, the students will examine how Newton’s Method works
and will use it to find the roots of a few equations. Their assignment will consist of applying
Newton’s Method to several problems and using derivatives to draw the complete graph of f(x) =
x3 – x. The students will also be instructed to find the roots of this problem algebraically without
using Newton’s Method.
On the second day, the students will draw on their experiences to write a graphing
calculator program that may be used to calculate Newton’s Method. A sample program is
provided in Appendix C. This day will be challenging. Even with extra copies of the Seirpinski
Gasket program on hand, the students will not have had experience writing programs that allow
the user to input information. Moreover, they will not know how to accommodate the number of
33
iterations chosen within the program. The best-case scenario has the students brainstorming all
the important features of the program with the teacher only contributing the appropriate
calculator commands. At the conclusion of class, the students will be split into groups and will
be told to investigate the included table using the new program. They will be finding the zero to
which each value on the table converges.
On day three, the groups will complete the worksheet that has a number line at the top.
The students will first use their findings about where convergence intervals begin and end to
place marks on the number line before finishing the page. A discussion of this activity will take
place at the end of the period or the start of day four.
Connections to Research
The students will be implementing algorithms in the traditional sense by using Newton’s
Method. Research agrees the use of algorithms in this manner is a continued necessity. The
students will also move beyond the informal notation proposed for initial use by Hart into the
formal standard notation that is used in recursive processes like the Mandelbrot Set found in the
last section. As research suggests, multiple opportunities will be provided for students to transfer
between algebraic, numeric, and graphic representations, including the use of a number line, as
they explore Newton’s Method. Lastly, the use of programming will continue in an effort to
obtain the positive side effects of problem translation and procedure comprehension documented
by Mayer, Dyck and Vilbers.
Future
The computer-generated pictures illustrating the basins of attraction of complex number
equations may provide an intriguing starting point for a new investigation. Newton’s Method
may be used to find the complex zeros, although, it is a little unusual at the high school level.
Nevertheless, it would make a solid connection to fractals.
34
Logistic Function – Pre-Calculus – Appendix D
Students are expected to have a working knowledge of various families of functions and
their characteristics prior to beginning this activity. They must be familiar with the term end
behavior and be experienced with a graphing calculator. Since this does not take place until
chapter four, students will have spent much of the first semester reviewing how to solve and
graph functions of various degrees including exponential and logarithmic functions. This is the
foundation on which the last activity will be developed.
This activity, in which population growth will be modeled using a logistic function, is
meant to supplement a very traditional pre-calculus textbook in which little development of the
logistic function is provided prior to the involvement of exponential functions. This activity will
start with a discussion of several possible models that may be used to represent population
growth. After the choices are narrowed to the logistic function, the students will experiment
with the model both numerically and graphically to determine whether it will work. They will
then use graphing calculators to investigate the effect of different growth rates on the model,
eventually resulting in chaos.
The first part of the activity, up to problem 13, will take place over the span of two days.
At this point, the completion of Part 2 Cobwebbing and Part 3 Feigenbaum Bifurcation Diagram
is unlikely. The cobwebbing lesson would provide a connection to the cyclic behavior of the
Mandelbrot Set. In a like manner, the bifurcation diagram would be the same one used in the
Mandelbrot discussion. The three lessons were designed to reinforce each other through the use
of multiple representations. However, the implementation of the last two is doubtful due to this
educator’s recent experience in the introduction of parametric graphing and many students’
unwillingness to work on their own.
35
Connections to Research
In choosing the logistic function to model population growth, the students will be forced
to use algorithmic thinking. They will also have the opportunity to gain experience using formal
recursive notation in the investigation of their model. Meanwhile, students will have multiple
chances to switch between numerical and graphical representations in their pursuit to understand
functions. Most of the suggested algebraic representation, nonetheless, will take place in the
days following this activity.
Future
Unless another pleasant surprise occurs, such as the ease of programming in the first
activity, the future will include the implementation of the cobwebbing and bifurcation lessons.
They will more than likely need to be adjusted to accommodate the new textbook series that is
being adopted. However, the mathematics will remain the same.
Conclusion
In reality, the part of this project that is the most exciting is the Mandelbrot Set. If
possible, that is the lesson this author would like to implement in her classroom. The
Mandelbrot Set provides much of what is missing in secondary school mathematics beginning
with the fact that it is not a thousand years old. It was strangely invigorating to discover a new
piece of mathematics that is younger than this educator while is capable of being taught at the
high school level. For a change, mathematics is not about Newton, the Greeks, or Euclid.
The use of the Mandelbrot Set could also answer one of the most common criticisms
leveled at secondary mathematics. That is the curriculum is a mile wide and an inch deep.
Perhaps, the best example of this lack of depth is the study of the complex number system.
Many students leave algebra two knowing they have studied imaginary numbers. Maybe, they
36
actual remember i2 = -1. For the most part, however, students do not understand the complex
number system.
This educator informally introduced a pre-calculus and a calculus class to the Mandelbrot
Set. In pre-calculus, it was used to illustrate the composition of functions. In calculus, its colors
were used to discuss the rate of change. Both groups were surprised to find out that imaginary
numbers really are used for something. Furthermore, the Mandelbrot Set earned one of the most
coveted but seldom heard comments in mathematics, “Neat!”.
The relationship between the Mandelbrot Set and secondary mathematics has been
established beyond the complex number system. The M-Set is a fractal. It is connected to the
logistics function, Newton’s Method, and chaos. Moreover, its study involves the application of
algorithmic and recursive thinking while transferring between algebraic, numeric, and graphical
representations. The Mandelbrot Set could easily become the foundation of a new secondary
mathematics class offered after algebra two for students who do not want to pursue the
pre-calculus/calculus track.
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 0.499959971 y24970 0 0.499959971 x24971 0.499959973 y24971 0 0.499959973 x24972 0.499959974 y24972 0 0.499959974 x24973 0.499959976 y24973 0 0.499959976 x24974 0.499959977 y24974 0 0.499959977 x24975 0.499959979 y24975 0 0.499959979 x24976 0.499959981 y24976 0 0.499959981 x24977 0.499959982 y24977 0 0.499959982 x24978 0.499959984 y24978 0 0.499959984 x24979 0.499959985 y24979 0 0.499959985 x24980 0.499959987 y24980 0 0.499959987 x24981 0.499959989 y24981 0 0.499959989 x24982 0.49995999 y24982 0 0.49995999 x24983 0.499959992 y24983 0 0.499959992 x24984 0.499959993 y24984 0 0.499959993 x24985 0.499959995 y24985 0 0.499959995 x24986 0.499959997 y24986 0 0.499959997 x24987 0.499959998 y24987 0 0.499959998 x24988 0.49996 y24988 0 0.49996 x24989 0.499960001 y24989 0 0.499960001 x24990 0.499960003 y24990 0 0.499960003 x24991 0.499960005 y24991 0 0.499960005 x24992 0.499960006 y24992 0 0.499960006 x24993 0.499960008 y24993 0 0.499960008 x24994 0.499960009 y24994 0 0.499960009 x24995 0.499960011 y24995 0 0.499960011 x24996 0.499960013 y24996 0 0.499960013 x24997 0.499960014 y24997 0 0.499960014 x24998 0.499960016 y24998 0 0.499960016 x24999 0.499960017 y24999 0 0.499960017 x25000 0.499960019 y25000 0 0.499960019
39
The convergence of B is a 2-cycle where (a,b) = (-1,0).
xn+1 = xn 2 – yn
2 + a = xn 2 – 02 -1 = xn
2 -1
(-1,0) converges to a 2-cycle.
xn+2 = xn and yn+2 = yn
xn+1 = xn 2 -1
xn+2 = (xn 2 –1)2 – 1
xn = (xn
- 1
Using TI-89, xn = 0, -1, 1.618033989, -.618033988
Attracts if | derivative at that point | ≤ 1. Repels if | derivative at that point | > 1.
xn+2 = (xn 2 –1)2 – 1
Let g(x) = (x2 –1)2 – 1
g’(x) = 2(x2 –1) · 2x = 4x(x2 –1) = 4x3 - 4x
g’ (0) = 0 attracts
g’ (-1) = 0 attracts
g’ (1.61803398875) = 10.472135955 repels
g’ (-.618033988) = 1.527864045 repels
Since 0 and –1 attract, this means that they will continue to cycle on the next page. The two
solutions that repel will continue to diverge from the fixed point.
40
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 0 y24970 0 0 x24971 -1 y24971 0 1 x24972 0 y24972 0 0 x24973 -1 y24973 0 1 x24974 0 y24974 0 0 x24975 -1 y24975 0 1 x24976 0 y24976 0 0 x24977 -1 y24977 0 1 x24978 0 y24978 0 0 x24979 -1 y24979 0 1 x24980 0 y24980 0 0 x24981 -1 y24981 0 1 x24982 0 y24982 0 0 x24983 -1 y24983 0 1 x24984 0 y24984 0 0 x24985 -1 y24985 0 1 x24986 0 y24986 0 0 x24987 -1 y24987 0 1 x24988 0 y24988 0 0 x24989 -1 y24989 0 1 x24990 0 y24990 0 0 x24991 -1 y24991 0 1 x24992 0 y24992 0 0 x24993 -1 y24993 0 1 x24994 0 y24994 0 0 x24995 -1 y24995 0 1 x24996 0 y24996 0 0 x24997 -1 y24997 0 1 x24998 0 y24998 0 0 x24999 -1 y24999 0 1 x25000 0 y25000 0 0
41
The convergence of C is a 3-cycle where (a,b) = (-1.75,0).
xn+1 = xn 2 – yn
2 + a = xn 2 – 02 -1.75 = xn
2 –1.75
(-1.75,0) converges to a 3-cycle.
xn+3 = xn and yn+3 = yn
xn+1 = xn 2 –1.75
xn+2 = (xn 2 –1.75)2 – 1.75
xn+3 = ((xn 2 – 1.75)2 –1.75)2 – 1.75
xn = ((xn
- 1.75
1.30193530428, -.054958148188, -.914213562373, -1.7462094325
Attracts if | derivative at that point | ≤ 1. Repels if | derivative at that point | > 1.
xn+3 = ((xn 2 – 1.75)2 –1.75)2 – 1.75 = ((xn
4 - 7/2xn 2 + 49/4) – 7/4)2 – 7/4
= (xn 4 - 7/2xn
6 + (119/8)xn 4 – (147/16)xn
h’ (x) = 8x7 - 42x5 + (119/2)x3 – (147/8)x
h’ (1.30193794065) = .999990485486 attracts
h’ (-.054958073375) = .999998952702 attracts
h’ (-1.74731452869) = .945556173139 attracts
h’ (1.91421356237) = 56.1126983705 repels
h’ (1.301935304288) = 1.00011293748 repels
h’ (-.054958148188) = 1.0000002872 repels
h’ (-.914213562373) = -6.1126983722 repels
h’ (-1.7462094325) = 1.12450973746 repels
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 -1.746981082 y24970 0 1.746981082 x24971 1.301942902 y24971 0 1.301942902 x24972 -0.05494468 y24972 0 0.05494468 x24973 -1.746981082 y24973 0 1.746981082 x24974 1.301942901 y24974 0 1.301942901 x24975 -0.054944682 y24975 0 0.054944682 x24976 -1.746981082 y24976 0 1.746981082 x24977 1.301942901 y24977 0 1.301942901 x24978 -0.054944684 y24978 0 0.054944684 x24979 -1.746981082 y24979 0 1.746981082 x24980 1.3019429 y24980 0 1.3019429 x24981 -0.054944685 y24981 0 0.054944685 x24982 -1.746981082 y24982 0 1.746981082 x24983 1.301942899 y24983 0 1.301942899 x24984 -0.054944687 y24984 0 0.054944687 x24985 -1.746981081 y24985 0 1.746981081 x24986 1.301942899 y24986 0 1.301942899 x24987 -0.054944688 y24987 0 0.054944688 x24988 -1.746981081 y24988 0 1.746981081 x24989 1.301942898 y24989 0 1.301942898 x24990 -0.05494469 y24990 0 0.05494469 x24991 -1.746981081 y24991 0 1.746981081 x24992 1.301942897 y24992 0 1.301942897 x24993 -0.054944692 y24993 0 0.054944692 x24994 -1.746981081 y24994 0 1.746981081 x24995 1.301942897 y24995 0 1.301942897 x24996 -0.054944693 y24996 0 0.054944693 x24997 -1.746981081 y24997 0 1.746981081 x24998 1.301942896 y24998 0 1.301942896 x24999 -0.054944695 y24999 0 0.054944695 x25000 -1.74698108 y25000 0 1.74698108
43
The convergence of G is a 4-cycle where (a,b) = (-1.3,0).
xn+1 = xn 2 – yn
2 + a = xn 2 – 02 -1.3 = xn
2 – 1.3
(-1.3,0) converges to a 4-cycle.
xn+4 = xn and yn+4 = yn
xn+1 = xn 2 –1.3
xn+2 = (xn 2 –1.3)2 – 1.3
xn+3 = ((xn 2 – 1.3)2 –1.3)2 – 1.3
xn+4 = (((xn 2 – 1.3)2 – 1.3)2 –1.3)2 – 1.3
xn = (((xn
- 1.3
1.7449899598, .24161984871, -.744989959799, -1.24161984867
Attracts if | derivative at that point | ≤ 1. Repels if | derivative at that point | > 1.
xn+4 = (((xn 2 – 1.3)2 – 1.3)2 –1.3)2 – 1.3 = (((xn
4 – 2.6xn
= ((xn 4
8 –5.2xn 6 + 7.54xn
p(x) = (x8 –5.2x6 + 7.54x4 –2.028 x2 –1.1479)2 – 1.3
p’(x) = 2(x8 –5.2x6 + 7.54x4 –2.028 x2 –1.1479)(8x7 –31.2x5 +30.16x3 –4.056x)
p’ (.389018548257) = .180542753191 attracts
p’ (.019430292333) = .180542753188 attracts
p’ (-1.14866456912) = .180542753361 attracts
p’ (-1.29962246374) = .180542753185 attracts
p’ (1.7449899598) = 148.35142169 repels
p’ (-1.24161984867) = 1.44000000013 repels
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 0.389018548 y24970 0 0.389018548 x24971 -1.148664569 y24971 0 1.148664569 x24972 0.019430292 y24972 0 0.019430292 x24973 -1.299622464 y24973 0 1.299622464 x24974 0.389018548 y24974 0 0.389018548 x24975 -1.148664569 y24975 0 1.148664569 x24976 0.019430292 y24976 0 0.019430292 x24977 -1.299622464 y24977 0 1.299622464 x24978 0.389018548 y24978 0 0.389018548 x24979 -1.148664569 y24979 0 1.148664569 x24980 0.019430292 y24980 0 0.019430292 x24981 -1.299622464 y24981 0 1.299622464 x24982 0.389018548 y24982 0 0.389018548 x24983 -1.148664569 y24983 0 1.148664569 x24984 0.019430292 y24984 0 0.019430292 x24985 -1.299622464 y24985 0 1.299622464 x24986 0.389018548 y24986 0 0.389018548 x24987 -1.148664569 y24987 0 1.148664569 x24988 0.019430292 y24988 0 0.019430292 x24989 -1.299622464 y24989 0 1.299622464 x24990 0.389018548 y24990 0 0.389018548 x24991 -1.148664569 y24991 0 1.148664569 x24992 0.019430292 y24992 0 0.019430292 x24993 -1.299622464 y24993 0 1.299622464 x24994 0.389018548 y24994 0 0.389018548 x24995 -1.148664569 y24995 0 1.148664569 x24996 0.019430292 y24996 0 0.019430292 x24997 -1.299622464 y24997 0 1.299622464 x24998 0.389018548 y24998 0 0.389018548 x24999 -1.148664569 y24999 0 1.148664569 x25000 0.019430292 y25000 0 0.019430292
45
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 -0.13021853 y24970 -0.7402484 0.751614663 x24971 -0.661010871 y24971 -0.5472119 0.858123656 x24972 0.007494534 y24972 -0.016574 0.018189711 x24973 -0.13021853 y24973 -0.7402484 0.751614663 x24974 -0.661010871 y24974 -0.5472119 0.858123656 x24975 0.007494534 y24975 -0.016574 0.018189711 x24976 -0.13021853 y24976 -0.7402484 0.751614663 x24977 -0.661010871 y24977 -0.5472119 0.858123656 x24978 0.007494534 y24978 -0.016574 0.018189711 x24979 -0.13021853 y24979 -0.7402484 0.751614663 x24980 -0.661010871 y24980 -0.5472119 0.858123656 x24981 0.007494534 y24981 -0.016574 0.018189711 x24982 -0.13021853 y24982 -0.7402484 0.751614663 x24983 -0.661010871 y24983 -0.5472119 0.858123656 x24984 0.007494534 y24984 -0.016574 0.018189711 x24985 -0.13021853 y24985 -0.7402484 0.751614663 x24986 -0.661010871 y24986 -0.5472119 0.858123656 x24987 0.007494534 y24987 -0.016574 0.018189711 x24988 -0.13021853 y24988 -0.7402484 0.751614663 x24989 -0.661010871 y24989 -0.5472119 0.858123656 x24990 0.007494534 y24990 -0.016574 0.018189711 x24991 -0.13021853 y24991 -0.7402484 0.751614663 x24992 -0.661010871 y24992 -0.5472119 0.858123656 x24993 0.007494534 y24993 -0.016574 0.018189711 x24994 -0.13021853 y24994 -0.7402484 0.751614663 x24995 -0.661010871 y24995 -0.5472119 0.858123656 x24996 0.007494534 y24996 -0.016574 0.018189711 x24997 -0.13021853 y24997 -0.7402484 0.751614663 x24998 -0.661010871 y24998 -0.5472119 0.858123656 x24999 0.007494534 y24999 -0.016574 0.018189711 x25000 -0.13021853 y25000 -0.7402484 0.751614663
46
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 0.059459847 y24970 -0.8171861 0.819346436 x24971 -0.394257635 y24971 -0.6271795 0.7408058 x24972 0.032084933 y24972 -0.0354594 0.047820602 x24973 0.269772076 y24973 -0.5322754 0.596736206 x24974 0.059459847 y24974 -0.8171861 0.819346436 x24975 -0.394257635 y24975 -0.6271795 0.7408058 x24976 0.032084933 y24976 -0.0354594 0.047820602 x24977 0.269772076 y24977 -0.5322754 0.596736206 x24978 0.059459847 y24978 -0.8171861 0.819346436 x24979 -0.394257635 y24979 -0.6271795 0.7408058 x24980 0.032084933 y24980 -0.0354594 0.047820602 x24981 0.269772076 y24981 -0.5322754 0.596736206 x24982 0.059459847 y24982 -0.8171861 0.819346436 x24983 -0.394257635 y24983 -0.6271795 0.7408058 x24984 0.032084933 y24984 -0.0354594 0.047820602 x24985 0.269772076 y24985 -0.5322754 0.596736206 x24986 0.059459847 y24986 -0.8171861 0.819346436 x24987 -0.394257635 y24987 -0.6271795 0.7408058 x24988 0.032084933 y24988 -0.0354594 0.047820602 x24989 0.269772076 y24989 -0.5322754 0.596736206 x24990 0.059459847 y24990 -0.8171861 0.819346436 x24991 -0.394257635 y24991 -0.6271795 0.7408058 x24992 0.032084933 y24992 -0.0354594 0.047820602 x24993 0.269772076 y24993 -0.5322754 0.596736206 x24994 0.059459847 y24994 -0.8171861 0.819346436 x24995 -0.394257635 y24995 -0.6271795 0.7408058 x24996 0.032084933 y24996 -0.0354594 0.047820602 x24997 0.269772076 y24997 -0.5322754 0.596736206 x24998 0.059459847 y24998 -0.8171861 0.819346436 x24999 -0.394257635 y24999 -0.6271795 0.7408058 x25000 0.032084933 y25000 -0.0354594 0.047820602
47
xn+1 = xn 2 - yn
2 + a yn+1 = 2xnyn + b (a2+b2)1/2
x24970 0.017015069 y24970 0.01479455 0.022547532 x24971 -0.509929366 y24971 0.56050346 0.757754635 x24972 -0.564136171 y24972 -0.0116343 0.564256127 x24973 -0.191885739 y24973 0.57312671 0.604395869 x24974 -0.801654093 y24974 0.34005031 0.870794753 x24975 0.017015069 y24975 0.01479455 0.022547532 x24976 -0.509929366 y24976 0.56050346 0.757754635 x24977 -0.564136171 y24977 -0.0116343 0.564256127 x24978 -0.191885739 y24978 0.57312671 0.604395869 x24979 -0.801654093 y24979 0.34005031 0.870794753 x24980 0.017015069 y24980 0.01479455 0.022547532 x24981 -0.509929366 y24981 0.56050346 0.757754635 x24982 -0.564136171 y24982 -0.0116343 0.564256127 x24983 -0.191885739 y24983 0.57312671 0.604395869 x24984 -0.801654093 y24984 0.34005031 0.870794753 x24985 0.017015069 y24985 0.01479455 0.022547532 x24986 -0.509929366 y24986 0.56050346 0.757754635 x24987 -0.564136171 y24987 -0.0116343 0.564256127 x24988 -0.191885739 y24988 0.57312671 0.604395869 x24989 -0.801654093 y24989 0.34005031 0.870794753 x24990 0.017015069 y24990 0.01479455 0.022547532 x24991 -0.509929366 y24991 0.56050346 0.757754635 x24992 -0.564136171 y24992 -0.0116343 0.564256127 x24993 -0.191885739 y24993 0.57312671 0.604395869 x24994 -0.801654093 y24994 0.34005031 0.870794753 x24995 0.017015069 y24995 0.01479455 0.022547532 x24996 -0.509929366 y24996 0.56050346 0.757754635 x24997 -0.564136171 y24997 -0.0116343 0.564256127 x24998 -0.191885739 y24998 0.57312671 0.604395869 x24999 -0.801654093 y24999 0.34005031 0.870794753 x25000 0.017015069 y25000 0.01479455 0.022547532
48
The programming code necessary to produce the Mandelbrot Set on a TI-89 is given
below. While many computer programs for the M-Set are available online, only one reference
for graphing calculators was found. The article was written for a TI-83. That program expects
the user to examine the M-Set one point at a time and to plot the results by hand. The TI-89
program that follows is set to increase by increments of two-hundredths and then to iterate 50
times before automatically plotting the point.
:mandelbr()
:-3.2→xmin r- maximum j value
:3.2→xmax p- maximum k value
:.1→xscl x- x value
:-1.5→ymin y- y value
:1.5→ymax nx- new x
:.1→yscl ny- new y
:Delar j,r,k,p,x,nx,ic,y,ny,d,lc,xlist,ylist,slist ic- iteration counter
:ClrIO d- distance squared
:2→r ylist- stores M-Set y values
:0→p slist- symmetry list for SPS
:0→lc
:nx→x
:ny→y
:Exit
:EndLoop
:EndIf
:EndFor
50
:EndFor
:EndPrgm
51
http://classes.yale.edu/Fractals/MandelSet/MandelGallery/Im35.jpg
52
Fractal Tree
1. At the bottom of this page, draw a vertical segment one unit long. 2. Draw two branches that are half the length of the last segment from the top of the
segment. 3. Continue repeating step 2 from the ends of your new branches until you reach 1/16.
54
4. Fill out the following table.
Number of Total Number Length of 1 Total Length Iteration New Branches of Branches New Branch of all Branches
0 1 2 3 4
N 5a) 5b) 5c) 5d)
5. What is the relationship between the number of iterations and the
a. number of new branches?
b. total number of branches?
c. length of 1 new branch?
d. total length of branches?
6. What happens to the length of a single new branch as n approaches infinity? Support your answer both numerically and graphically.
(Naylor, 1999)
1st iteration 2nd iteration 3rd iteration
1 The Cantor set is formed by taking the original segment (initiator) of length 1 unit and splitting it into thirds. The center third is then erased. Use this to fill in the following table. Length of each Number of Total Length of Iteration New Segment New Segments New Segments
0 1 2 3 4
n 2. What happens to the number of new segments as n approaches infinity? Support your answer both numerically and graphically.
3. What happens to the total length of new segments as n approaches infinity? Support your answer both numerically and graphically.
(Naylor, 1999; Lornell and Westerberg, 1999; http://spanky.triumf.ca/www/fractal-info/cantor.htm)
Sierpinski Gasket (Triangle) This fractal is starts with an equilateral triangle. We then find the midpoint of each side before connecting the all three of them to form another triangle. This center triangle is then removed.
TI-84 Sierpinski Triangle Program
:.5(.5+X) → X
:.5(1+X) →X
57
Use the result of running the Sierpinski’s program to help answer the following questions.
1. What happens to the area of the original triangle as the number K of iterations approaches 3000? Infinity? Explain. 2. How many points will never be removed?
3. How can the area approach ________________in #1 if ______________ points remain in #2?
(Naylor, 1999; http://www.fractovia.org/what/what_ing1.shtml, http://education.ti.com/us/product/tech/84pse/guide/84pseguideusprint.html, Bedford, 1998)
Calculus
64
Terms 1. derivative: rate of change or slope of a tangent line 2. linearization: process used to approximate a small portion of a complicated graph with a tangent line 3. maxima: all local and absolute maximums on a graph 4. minima: all local and absolute minimums on a graph 5. inflection point: the point(s) where the graph changes from concave up to concave down My calculus students like to think of it as the point that determines whether water will stay in the bowl or dump out of it. 6. zeros, roots, solutions: the point(s) where the graph crosses the x-axis
A - local minimum B - inflection point C - local maximum D - zero, root, x-intercept, solution E – tangent line through point C to the curve
65
Newton’s Method and Chaos This lesson takes place after Newton’s Method has been introduced and students have had the opportunity to use it to locate roots. The following problems were assigned as part of the homework for today.
1. Show a complete graph and identify the inflection points, local maximum and minimum values, and the intervals on which the graph is rising, falling, concave up, and concave down. Do analytically, then confirm graphically.
f(x) = x3-x
local maximum: (-√3/3, .807550)
local minimum: (√3/3, -.384900)
falling: [-√3/3,√3/3]
concave up: (0,∞)
concave down: (-∞,0)
2. Find the roots of the above problem algebraically. Do not use Newton’s Method.
Answers: -1,0,1
66
Newton’s Method and Chaos (in class) We are going to investigate f(x) = x3-x using Newton’s Method. However, to do it more easily, we are going to write a program for our calculator. Students’ program for Newton’s should be similar to below. xn = xn-1 – f(xn-1)/f’(xn-1) Program: NEWTONS :ClrDraw :PlotsOff :Disp “GUESS FOR ROOT” :Input X :Disp “NUMBER ITERATIONS” :Input I :0→C :Lbl 1 :(X-(X^3-X)/(3X^2-1))→X ******SEE NOTE BELOW****** :Disp X :C+1→C :If C<I :Then :Goto 1 :End ***Storing f(x) under Y1 allows you to use the following to replace above step*** (X-Y1/nDeriv(Y1,X,X))→X This step should error out on the critical points. It won’t if you use nDeriv.
67
Investigate f(x) = x3-x using the Newton’s Method program. Write down whether each value converges to –1,0, or 1. Then, find the exact point where each interval begins and ends to four decimal place. Note: You do not need to try every value below. The table is given to help organize your thoughts. Use the blank Extras column to record your interval investigations.
-1.1 -0.66 -0.22 0.22 0.66 Extras -1.09 -0.65 -0.21 0.23 0.67 -1.08 -0.64 -0.2 0.24 0.68 -1.07 -0.63 -0.19 0.25 0.69 -1.06 -0.62 -0.18 0.26 0.7 -1.05 -0.61 -0.17 0.27 0.71 -1.04 -0.6 -0.16 0.28 0.72 -1.03 -0.59 -0.15 0.29 0.73 -1.02 -0.58 -0.14 0.3 0.74 -1.01 -0.57 -0.13 0.31 0.75
-1 -0.56 -0.12 0.32 0.76 -0.99 -0.55 -0.11 0.33 0.77 -0.98 -0.54 -0.1 0.34 0.78 -0.97 -0.53 -0.09 0.35 0.79 -0.96 -0.52 -0.08 0.36 0.8 -0.95 -0.51 -0.07 0.37 0.81 -0.94 -0.5 -0.06 0.38 0.82 -0.93 -0.49 -0.05 0.39 0.83 -0.92 -0.48 -0.04 0.4 0.84 -0.91 -0.47 -0.03 0.41 0.85 -0.9 -0.46 -0.02 0.42 0.86
-0.89 -0.45 -0.01 0.43 0.87 -0.88 -0.44 0 0.44 0.88 -0.87 -0.43 0.01 0.45 0.89 -0.86 -0.42 0.02 0.46 0.9 -0.85 -0.41 0.03 0.47 0.91 -0.84 -0.4 0.04 0.48 0.92 -0.83 -0.39 0.05 0.49 0.93 -0.82 -0.38 0.06 0.5 0.94 -0.81 -0.37 0.07 0.51 0.95 -0.8 -0.36 0.08 0.52 0.96
-0.79 -0.35 0.09 0.53 0.97 -0.78 -0.34 0.1 0.54 0.98 -0.77 -0.33 0.11 0.55 0.99 -0.76 -0.32 0.12 0.56 1 -0.75 -0.31 0.13 0.57 1.01 -0.74 -0.3 0.14 0.58 1.02 -0.73 -0.29 0.15 0.59 1.03 -0.72 -0.28 0.16 0.6 1.04 -0.71 -0.27 0.17 0.61 1.05 -0.7 -0.26 0.18 0.62 1.06
-0.69 -0.25 0.19 0.63 1.07 -0.68 -0.24 0.2 0.64 1.08 -0.67 -0.23 0.21 0.65 1.09
68
-1 0 0 1 1. Label the local minimum, maximum, inflection point, and roots on the number line. 2. Color the intervals that converge to
a) –1 ___________________ (Choose and color the number line.) b) 0 ___________________ c) 1 ___________________
3. What is one characteristic of fractals that is present? 4. What does chaos mean to you? What do you think chaos means in math? Put rectangles around the regions on the number line that represent chaos.
(Fischer, 2001; Vanden Bosch, 2000; Bedford, 1998)
69
Pre-Calculus
70
1. Are the following models realistic interpretations of population growth? Explain. Let a = rate of growth.
a) Linear Growth Model y = ax Let a = 2.5
[-10,10] by [-10,10]
b) Exponential Growth Model y = ax Let a = 2.5
[-10,10] by [-5,15] 2. What factors should you consider when trying to predict population growth? 3. Would the following model be a better representation of population growth? Explain.
Logistic Growth Model
71
5. Use Pn+1 = aPn(1-Pn) to find the population for the first 15 generations. Let a = 2.5 and Pn = .02
P0+1 = P1 =
P1+1 = P2 =
P2+1 = P3 =
P3+1 = P4 =
P4+1 = P5 =
P5+1 = P6 =
P6+1 = P7 =
P7+1 = P8 =
P8+1 = P9 =
P9+1 = P10 =
P10+1 = P11 =
P11+1 = P12 =
P12+1 = P13 =
P13+1 = P14 =
P14+1 = P15 = 6. Plot the 15 ordered pairs you just found (n+1, Pn+1) n+1 = generation # Example: 0+1 = 1st generation
Pn+1 n+1
7. What does the answer to #5 and the end behavior asymptote on #6 seem to conclude?
72
Part 1 Next, we are going use your graphing calculator to investigate what happens if we change the growth rate. We will use the same logistic function above in the sequence mode of your calculator.
Pn+1 = aPn(1-Pn) which is equivalent to y = ax(1-x) Calculator notation u(n) = Au(n-1)(1-u(n-1))
TI-83 and 84 Note: u is obtained by pressing 2nd and 7
Mode Y =
nMin and nMax tells it to evaluate 1 through 50
Plot Start = first number to be plotted Plot Step = tells increment by which Plot Start increases 8. Return to home screen and store 2.9 as A then graph. Sketch of graph Table
How does this compare to your answer in number 7?
Explain. Store the following values for A, sketch a graph and list the table’s values 45≤n≤51. 9. A = 3.3 Sketch of graph Table Describe the end behavior of this sequence.
73
10. A = 3.54 Sketch of graph Table Describe the end behavior of this sequence. Another sketch in connected mode 11. A = 3.9 Sketch of graph Table Describe the end behavior of this sequence. 12. What is chaos? 13. What should this mean to everyone who uses mathematical modeling to study real
world issues?
ANSWER KEY – Part 1
Next, we are going use your graphing calculator to investigate what happens if we change the growth rate. We will use the same logistic function above in the sequence mode of your calculator.
Pn+1 = aPn(1-Pn) which is equivalent to y = ax(1-x) Calculator notation u(n) = Au(n-1)(1-u(n-1))
TI-83 and 84 Note: u is obtained by pressing 2nd and 7
Mode Y =
nMin and nMax tells it to evaluate 1 through 50
Plot Start = first number to be plotted Plot Step = tells increment by which Plot Start increases 8. Return to home screen and store 2.9 as A then graph. Sketch of graph Table
How does this compare to your answer in number 7?
Explain. Store the following values for A, sketch a graph and list the table’s values 45≤n≤51. 9. A = 3.3 Sketch of graph Table Describe the end behavior of this sequence.
End behavior is two valued.
"Bifurcation"
75
ANSWER KEY CONTINUED – Part 1 10. A = 3.54 Sketch of graph Table Describe the end behavior of this sequence.
End behavior is four valued.
Have students change to connected mode to see oscillation. connected 11. A = 3.9 Sketch of graph Table Describe the end behavior of this sequence.
Fluctuates randomly Chaos 12. What is chaos? “property of a mathematical system where a small difference in initial conditions could result in radically different predictions” 13. What should this mean to everyone who uses mathematical modeling to study real
world issues?
76
Part 2 Cobwebbing TI-83 and 84 Note: u is obtained by pressing 2nd and 7
2nd Format
Compare this to #8 in Part 1. You will see
y = ax(1-x) Every two presses of right arrow y = x is 1 more step of iteration. 15. Store A = 3.3 Graph, Trace, and Sketch What happens? Compare this to #9 in Part 1.
77
16. Store A = 3.54 Graph, Trace, and Sketch What happens? Compare this to #10 in Part 1. 17. Store A = 3.9 Graph, Trace, and Sketch What happens? Compare this to #11 in Part 1. 18. What is the relationship between the activity in Part 1 and this activity in Part 2?
78
Part 2 – MY ANSWER KEY Cobwebbing TI-83 and 84 Note: u is obtained by pressing 2nd and 7
2nd Format
Compare this to #8 in Part 1. You will see
y = ax(1-x) Every two presses of right arrow y = x is 1 more step of iteration.
converges to 1 value 15. Store A = 3.3 Graph, Trace, and Sketch What happens? Compare this to #9 in Part 1.
Two valued Rotates Through Two Corners 16.
79
Store A = 3.54 Graph, Trace, and Sketch What happens? Compare this to #10 in Part 1.
"Four valued" - rotates through corners 17. Store A = 3.9 Graph, Trace, and Sketch What happens? Compare this to #11 in Part 1.
chaos - cobwebs everywhere 18. What is the relationship between the activity in Part 1 and this activity in Part 2?
80
PRGM
Use Catalog (2nd 0) to find FnOff and other keys. Note: Window features are found under Vars, 1 Window
To run the program, 2nd Quit, PRGM, EXEC, Enter. You should obtain a diagram that looks like the following. 20a. Where is there only one branch in the bifurcation diagram? Compare this to 8 and 14. 20b. When does A converge to a single point? 21a. Where are there two branches in the bifurcation diagram? Compare this to 9 and 15.
81
21b. When is A two-valued? 22a. Where are there four branches in the bifurcation diagram? Compare this to 10 and 16. 22b. When is A four-valued? 23a. Where does chaos reign in the bifurcation diagram? Compare this to 11 and 17. 23b. Which A values produce chaos? (Hanna, 2001)
82
PRGM
Use Catalog (2nd 0) to find FnOff and other keys. Note: Window features are found under Vars, 1 Window
To run the program, 2nd Quit, PRGM, EXEC, Enter. You should obtain a diagram that looks like the following. 20a. Where is there only one branch in the bifurcation diagram? Compare this to 8 and 14. 20b. When does A converge to a single point? 1<A<3 21a. Where are there two branches in the bifurcation diagram? Compare this to 9 and 15.
83
21b. When is A two-valued? 3<A<3.4 22a. Where are there four branches in the bifurcation diagram? Compare this to 10 and 16. 22b. When is A four-valued? 3.4<A<3.57 23a. Where does chaos reign in the bifurcation diagram? Compare this to 11 and 17. 23b. Which A values produce chaos? A≥3.57 *** Show region of stability in midst of chaos – 3-cycle. (Hanna, 2001)
84
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Feigenbaum Constant
Part 3
Connections to Research
Connections to Research
Population Growth Modeled with Logistic Functions
TI-83 and 84 Note: u is obtained by pressing 2nd and 7
ANSWER KEY – Part 1
TI-83 and 84 Note: u is obtained by pressing 2nd and 7
Part 2
Cobwebbing
TI-83 and 84 Note